Self-assembly of magnetic nanostructures

Printed on January 4, 1996

Self-assembly of magnetic nanostructures David Tomanek, Seong Gon Kim, Philippe Jund Department of Physics and Astronomy Michigan State University, East Lansing, Michigan 48824-1116, USA

Peter Borrmann, Heinrich Stamerjohanns, and Eberhard R. Hilf Department of Physics, University of Oldenburg D-26111 Oldenburg, Germany

(Received

)

Abstract We use Monte Carlo and quaternion molecular dynamics simulations to study the self-assembly of intriguing structures which form in colloidal suspensions of small magnetite particles. We show that the only stable isomers with few particles, a ring and a chain, can be eciently interconverted using a magnetizable tip. We propose to use the oscillating dipole eld of the tip to locally anneal the aggregates to either a ring in zero eld or a chain in nonzero applied eld.

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Given the present advanced stage of miniaturization, the most promising way to signi cantly reduce the dimension of devices involves a transition from micro-manufacturing to self-assembly of nanostructures [1]. Inspired by the richness of structures observed in aggregates of magnetic nanoparticles [2,3] and the possibility of their structural transformation [4,5], we propose a hybrid thermodynamic self-assembly technique capable of producing magnetic patterns of unprecedented packing density [6]. The key ingredients are a system of magnetic nanoparticles in a colloidal suspension, resonant magnetic heating on the nanometer scale that we postulate, and the possibility to manipulate individual nanostructures using a local magnetic eld. In the following, we prove our technique to work using realistic Monte Carlo and Molecular Dynamics simulations addressing the self-assembly, the eld-assisted interconversion, and the long-term stability of the magnetic nanostructures. In the following, we will describe microscopically the structural and magnetic transitions in microcanonical and canonical ensembles of few magnetic particles. Commercially available spherical nanoparticles of magnetite are covered by a thin surfactant layer to inhibit irreversible coalescence in a viscous liquid at room temperature [2]. Such colloidal suspensions, called ferro uids, have recently become a focus of experimental and theoretical attention due to their interesting behavior in applied magnetic elds [2,3,7{12]. We will discuss the eect of eld and temperature on the stability of the individual isomers, which { for few particles { are known to be either a chain or a ring [4]. More important, we will show how to locally modify their equilibrium structure by changing the eld and temperature (assembly of nanostructures) and how to distinguish magnetically between the dierent isomers (detection of nanostructures). Our model system consists of six [13] spherical magnetite particles with a diameter  = 200  A, mass m = 1:31107 amu, inertia I = 5:25107 amu A2, and a large permanent magnetic moment 0 = 1:68105 B . The potential energy of this system in an external magnetic eld B~ ext consists of the interaction between each particle i and the applied eld, given by ui = ?~i
B~ ext, and an interaction between all pairs of particles i and j (which are separated by ~rij = ~rj ? ~ri and carry the magnetic moment ~i = 0^i [14]), given by [4] 2

uij = (20=rij3 ) [^i
^j ? 3(^i
r^ij )(^j
r^ij )] ! !# " r ?  r ?  ij ij ? exp ?  : + exp ?  1 2

(1)

The rst term in this expression describes a dipole-dipole interaction. The second term describes a soft-core repulsion between the particles, mediated by a surfactant layer, and a long-range van der Waals attraction, partially screened by the suspending medium. We use 1 = 5:0  A, 2 = 21 = 10:0  A and  = 64  10?3 eV to describe a realistic ferro uid. This choice of parameters yields a potential with a strongly repulsive core and a weakly < 10% of the typical dipole-dipole interaction) \skin" around the particles with attractive ( < 40  a short range of  A. Depending on the magnetic eld, the equilibrium geometry of this system at low temperatures is either a ring with zero total magnetic moment or a chain with the magnetic moment  = max = N0 . As illustrated in the inset of Fig. 1(a), rings are more stable in zero eld, whereas chains are more stable in high magnetic elds Bext [4]. The large minimum potential energy barrier per particle
E  0:16 eV, corresponding to a \melting" temperature TM  630 K, prevents metastable chains in zero eld from closing to rings at room temperature. On the other hand, rings do not fragment into chains, unless exposed to high magnetic elds Bext >  600 Gauss [4,5], and hence are not disturbed by the low elds generated by aggregates in neighboring cells. This establishes the required stability of the magnetic structure [15]. Next, we studied the eciency of the eld-assisted assembly process. Results of a room temperature Monte Carlo simulation in applied elds Bext = 0 and Bext = 100 Gauss are presented in Fig. 1(a). These data indicate that upon applying a high magnetic eld for suciently long time, the majority of the systems will form a chain. In absence of a eld, after careful annealing, the majority of the systems will form a ring. Both isomers can be easily distinguished by separate peaks in the distribution of magnetic moments. Consequently, we will use the magnetic moment as the single characteristic of the nanostructure. In order to estimate the time needed to assemble a nanostructure, we performed Molec3

ular Dynamics simulations of the transition between a ring and a chain in a microcanonical ensemble of six magnetite particles. We made use of the quaternion formalism [16{18] to avoid divergencies in the orientational equations of motion which would otherwise occur in this system of magnetic spherical tops (with a nonvanishing mass and inertia) due to discontinuities in Euler angle coordinates. We used time steps
t = 510?11 s and integrated the equations of motion numerically using a fourth-order Runge-Kutta algorithm, since this method proved to be more stable and to better conserve the energy than alternate integration schemes. These and our above Monte Carlo studies suggest that heating up the system 100 K above room temperature reduces the average time for a structural transformation by one order of magnitude and hence signi cantly accelerates the assembly. On the other hand, the higher vibrational entropy of the chain in zero eld (as compared to the ring) plays an increasingly important role at these higher temperatures. This has no adverse eect on the ring-to-chain conversion in nonzero elds, but reduces the fraction of rings in zero eld and hence the eciency of the chain-to-ring conversion. The feasibility of a high packing density of nanostructures depends on the availability of an extremely localized source of magnetic eld and heat. As a promising technical realization, we suggest to use a soft magnetic nanotip, surrounded by a coil, as the source of localized static and oscillating magnetic eld. This nanoscopic electromagnet assembly can be suspended on a cantilever using the technology developed for the Atomic Force Microscope (AFM) [19]. The capability to assist in the assembly and detection of magnetic nanostructures with a precision of 100 ? 1000  A might be relatively simple to achieve in view of the AFM's success to obtain atomic resolution [19]. For eld assisted assembly, a sharp magnetic tip has several advantages. (i) The eld inhomogeneity guarantees that neighboring structures are not disturbed and that magnetite particles aggregate faster in the tip region. (ii) The tip can be used to generate a locally large static eld to assemble a chain. (iii) Fast eld reversal can be used to detach any aggregate from the tip. (iv) An oscillating high-frequency eld, generated by the tip, can be 4

used to excite preferentially the transverse bending modes of the chain, hence accelerating ring closure in a cooling environment [20]. The sharp tip, suspended on the cantilever of a Magnetic Force Microscope, can also be used to investigate the magnetic structures. The detection process is initiated by applying a weak inhomogeneous magnetic eld which will attract only magnetic aggregates (chains, but not nonmagnetic rings) to the tip. The presence of a chain attached to the tip will lead to a lowering of the mechanical resonance frequency of the cantilever-tip system that can be detected. This allows for a discrimination between a chain and a ring in a nondestructive way. We model the magnetic tip by a nonmagnetic cone with an opening angle of 60, which is rounded o at the end and terminated by a magnetizable sphere (see Fig. 2). The diameter of this sphere, tip = 400  A, is twice that of the magnetite particles in the colloidal suspension, and its magnetic moment is aligned with the cone axis. The nonmagnetic part of the interaction between the tip and the magnetite particles is assumed to be purely repulsive. In analogy to the corresponding term in Eq. (1), it is given by ur =  exp(?d=1), where d is the minimum distance between the surfaces of the tip and the magnetite particle in the colloid. The dynamics of the structural transformation, assisted by the local eld of a sharp magnetic tip, is illustrated in Fig. 1(b). To accelerate a ring-to-chain conversion, we rst heated the system locally by a high-frequency magnetic eld. We found that a dipole eld of frequency  = 1 MHz, generated by changing periodically the direction of the dipole moment at the tip tip = 7105 B , is most ecient in heating up the system and exciting the bending mode of the chain. The system, which had reached an average temperature of 500 K after 5 s, was subsequently cooled down during the next 5 s in the static eld of the tip dipole tip = 7105 B by extracting stepwise the energy doses of 15 meV, each followed by 1 s equilibration time. The same annealing schedule has been used for a chain-to-ring conversion, with the exception of using a smaller value for the oscillating tip dipole moment tip = 4105 B during the annealing process and tip = 0 during the cooling process. As 5

seen in Fig. 1(b), the average time needed to convert a ring to a chain or vice versa lies close to 5s [21]. The eld-assisted assembly process with a sharp magnetic tip is illustrated in Fig. 2 by snapshots from a Molecular Dynamics simulation. At the starting point of our simulation, shown in Fig. 2(a), the magnetite particles are randomly distributed and oriented in zero eld. The assembly of a chain is initiated by a static magnetization of the tip. This causes the particles to aggregate in the region of strongest Bext- eld and to form a chain aligned with the eld lines that is attached to the tip, as shown in Fig. 2(b). Subsequent reversal of the magnetization of the tip causes the chain to detach from the tip, as shown in Fig. 2(c). At this point, a stable chain is formed. As illustrated in Fig. 2(d), applying a high-frequency dipole eld excites the bending mode of the chain eciently, facilitating closure to a ring. Fig. 2(e) shows the spontaneous self-assembly of the ring structure after the eld had been switched o. The stability of this structure increases as it cools down in the suspending liquid. In conclusion, we proposed and modeled a hybrid self-assembly technique for aggregates consisting of magnetite nanoparticles, that is capable of producing magnetic patterns with unprecedented density. When viewed as information, this data density would by far exceed that of conventional magnetic and protein-based memories [22]. The key to tailored magnetic nanopatterns are the substantially dierent magnetic moments of the only stable isomers with few magnetite particles which are a ring and a chain. We proposed an ecient process to assemble and to detect individual nanostructures using the localized static and oscillating dipole eld of a sharp magnetic tip. We believe that the technique proposed here may bring us closer to nanopatterning on the atomic scale.

ACKNOWLEDGEMENTS DT, PJ and SGK acknowledge nancial support by the National Science Foundation under Grant Number PHY-92-24745 and the Oce of Naval Research under Grant Num6

ber N00014-90-J-1396. Our extensive computer simulations have been performed on the CRAY-T3D/192 of the Konrad-Zuse-Institute in Berlin and the S400 supercomputer of the Regionales Rechenzentrum fur Niedersachsen(RRZN) in Hannover.

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[13] The exact number of particles in the system is not critical, since all systems with